1990 AMC 12 Problem 27

Question

Which of these triples could not be the lengths of the three altitudes of a triangle?

1,

\sqrt3,

2

3,

4,

5

5,

12,

13

7,

8,

\sqrt{113}

8,

15,

17

Answer: C

Concepts:triangle inequality altitude reciprocal

Difficulty rating: 2380

Small hint:

For fixed area

K,

a side corresponding to altitude

h

equals

\frac{2K}{h}

Big hint:

Test the triangle inequality on the reciprocals of each triple

Solution:

If the altitudes are

h_1,h_2,h_3,

then the corresponding sides are proportional to

\frac{1}{h_1},\frac{1}{h_2},\frac{1}{h_3}.

For

5,12,13,

\frac15\gt\frac1{12}+\frac1{13},

so the reciprocals fail the triangle inequality. Direct checking shows that the reciprocals of each other listed triple satisfy all strict triangle inequalities.

Thus the correct answer is C.